Algebra Know-It-ALL

These values are all exact! But now suppose you want to compare the natural logs of these same five argu-

ments. The base-e logarithm of a number is the power of e that produces that number. You must use a

calculator to find natural logs. The results are as follows, rounded off to four decimal places in each case

(except the natural log of 1, which is an exact value):

ln 0.01 ≈−4.6052

ln 0.1 ≈−2.3026

ln 1 = 0

ln 10 ≈ 2.3026

ln 100 ≈ 4.6052

The log of 1 is always equal to 0, no matter what the base, because any positive real number raised to the

zeroth power is equal to 1.

How Logarithms Work

With logarithms, you can convert products into sums, ratios into differences, powers into

products, and roots into ratios.

Changing a product to a sum

Imagine two positive real number variables x and y. The logarithm of their product, no matter

what the base b happens to be (as long as b > 0), is always equal to the sum of the logarithms

of the individual numbers. You can write this as

logbxy= logbx+ logby

Let’s look at a numerical example. Consider the arguments exact. Use your calculator to fol-

low along:

log 10 (3 × 4) = log 10 3 + log 10 4

Working out both sides and approximating the results to four decimal places, you should get

log 10 12 ≈ 0.4771 + 0.6021

≈ 1.0792

You shouldn’t expect to get perfect answers every time you use logarithms, because the results

are almost always irrational numbers. That means they are endless, non-repeating decimals.

Approximation is the best you can do.

Changing a ratio to a difference

Again, suppose that x and y are positive real numbers. Then the logarithm of their ratio, regard-

less of the base b, is equal to the difference between the logarithms of the individual numbers:

logb (x/y)= logbx− logby

482 Logarithms and Exponentials